Please assist with this physics homework 4. Normalize the following wavefunctions(a) (x)=sin(); for a particle in a 1D box of length L.(b) (r) = xe-z/(c) (z) = (-²²)+(z)5. In a region of space, a particle with mass m and with zero energy has a time-independent wave-function (x) = Ae-2/2, where A and L are constants. Useyour knowledge of the Schrödinger equation to determine the potential energy V(x)of the particle. Plot the potential function? What is the minimum potential energyfor the particle, if it is an electron and L = 1 fm? Is this potential repulsive orattractive?6. Plot qualitative wavefunctions for the scenarios shown below.V=8ZTE√20VoJ.EVaVo VeoV=D1v(x)

Approach to solving the question:see the condition of normalization and by integration you can…

Answered: 4. Normalize the following…

Physics for Scientists and Engineers with Modern Physics

10th Edition

ISBN:9781337553292

Author:Raymond A. Serway, John W. Jewett

Publisher:Raymond A. Serway, John W. Jewett

Chapter39: Introduction To Quantum Physics

Section: Chapter Questions

Problem 26P

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An electron in a box is in the ground state with energy 2.0 eV. (a) Find the width of the box. (b) How much energy is needed to excite the electron to its first excited state? (c) If the electron makes a transition from an excited state to the ground state with the simultaneous emission of 30.0-eV photon, find the quantum number of the excited state?
1. Particle in a Box. A particle of mass m that is confined in a one- dimensional box of length L, i.e. x € (0, L), is described by the wave function: (x, t) = A sin where En = nnx L expli- n²π²ħ² 2m [², Ent ħ where n E N where n E N. The wave function is zero outside the box. Calculate the normalization constant A and compute the uncertainty of position and momentum regardless of the quantum number n.
6. An electron in hydrogen atom is in initial state Þ(r, 0) = A(2410o + iÞ210 + 421–1 – 2ib211) where wnim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? hw = - n2 c. Write the state Þ(r, t) at time t, using energy eigenvalues as En d. Find the expectation value of L in the state Þ(r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of Lz led to the value –ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities?
2. Suppose a particle of mass, m, has energy E, and wave function: WE (x, t = 0) = Aeikx + Be-ikx What is WE(x, t)? Calculate the probability density of the particle when it has the wavefunction, E (x, t). If you wish to simply your answer algebraically, use this information: let A = aeia and B = beif and A = a - B. where the variables a, b, a, ß are all real! WARNING: Simplifying the equation is somewhat time consuming. Show/explain why p(x,t) is not normalizable. According to quantum theory, all physical systems must have an associated wavefunction that is normalizable. Explain why plane wave solutions to Schrodinger's equation are used in quantum theory, despite not being normalizable?
Q3. Consider an infinite potential well of width d. In transitions between neighboring values of n, particles of mass that is in a position state as: |1 TX -iwot sin 2πχ sin d f(x. t) = + -iwit d (a) Proof that f(x.t) is still normalized for all value of t. (b) Find the probability distribution P(x.t) = |f(x.t)|²
Q3. Consider an infinite potential well of width d. In transitions between neighboring values of n, particles of mass that is in a position state as: 2πχ sin e-iwit d TX f(x. t) = -e-iwot + sin (a) Proof that f (x. t) is still normalized for all value of t. (b) Find the probability distribution P(x.t) = |f(x. t)|²
3. A harmonic oscillator of mass m and angular frequency w is in the initial state of wavefunction p(x,0) = Ai4o(x) + 2Ai¢2(x) Obtain the constant A b. Write the function (x, t) c. Calculate the uncertainties Ax and Ap in the state of wavefunction p(x,t) and show that the Heisenberg uncertainty principle is satisfied a.
8. If the wave function has the following equation y = 4cosA + 4isinA a. Determine the conjugate b. |y*y|
2. A particle is confined to the x-axis between x = 0 and x = 3a. The wave function of the particle is = Ax sin (5x). a. b. Determine (x) without determining A. Determine ² = ((x - (x))²) without determining A.
6. An electron in hydrogen atom is in initial state p(r, 0) = A(2µ100 + ¡Þ210 + 4Þ21–1 – 2i4211) where ynim are the eigenfunctions of the hydrogen atom a. Determine the constant A b. What is the probability of finding the electron in the first excited state? c. Write the state p(r, t) at time t, using energy eigenvalues as En = d. Find the expectation value of L in the state (r,t e. Find the expectation values of Lx and Ly in the state (r, t f. If measurement of L, led to the value -ħ what will be results of measurement of energy and the square of total orbital momentum immediately afterwards and what are their probabilities? hw n2
1. Consider a particle in one dimension with amplitude (x) given by V(x) = Aexp 242 Normalize this wavefunction. Hint: use this integral 00 -ax? dx a 2. Consider a particle of momentum p = 96hm-1. Here m stands for the unit meters. What is the probability density of finding it at the position x? 3. Write the wavefunction 6Tx v(x) = A cos as the linear combination of wave-functions of definite momenta (Hint: you need just two states of definite momentum). 4. Normalize the above wavefunction. 5. Plot the wavefunctions corresponding to the first three energy levels of a particle in a box.
01: The wave function for a particle in 1-D box of length (a) is Y -Asin (7x/a). Find the value of A for a box of length 200 nm. [ Marks 5 Q2: A particle of mass m moves in potential well of length 2L. Its potential energy is ( Marks 51 ) = (-) - { 7+ > >7- and described by the wave function V(x) * +L (-1) = A 7+ > *> 1- * +L Determine the Energy of the particle in terms of h.m and L Q3:A) The energy levels in this form E-! represents the nature of i) conductor 2 m L² [Marks 3] i) semiconductor üi) insulator. B) In finite potential barrier , how far a particle penetrate in to the barrier? What is the effect of j mass D energy, of the particle on penetration distance. [Marks 3] Q4: The energy of a particle in 2-D box is E- . Find the quantum numbers and the degree of 2ml degeneracy. [Marks 4] 05: The normalized wave function for a particle in l-D box is Y(x) - sin Find the average energy for this particle. [Marks 51

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